"Could you answer me?"

It should be accepted.

Then it would be better "не могли бы вы мне ответить?"

"Can you answer me?" - Ты можешь мне ответить? / Вы можете мне ответить?

Could you answer me? - Не мог бы ты мне ответить? / Не могли бы вы мне ответить?

I hope you see the difference.

"Can" is expressing an ability to answer. (Можешь/можете). "Could" is expressing politeness. (Мог бы/могли бы) Just another example of bad teaching from Duo. Please report it, may be some day it will be corrected.

Можешь ответить мне should be ok.

If you ask "можешь отвечать мне? " it will be like "are you physically able to answer me in general ? ". Or you may use it if you are going to ask a lot of questions and the person is going answering you during continuous period of time.

I believe your variant is correct too, you should report to the course maintainers it next time you get this sentence. The course is still in beta, so sometimes correct variants are not accepted.

Отвечать is imperfective, ответить is perfective.

Perfective verbs describes an action with reference to just one point of time. For ответить, this time is when the answer was successfully given.

Отвечать, on the other hand, describes an action as something that has a beginning and end, it emphasises process. You can do this either because the result is not achieved, or because it's less important than the process, or because the action is repeated. Imperfective verbs often correspond to continuous forms in English.

In this example, "Вы можете мне отвечать?" would be understood to mean a repeated action, i.e. it means something like "Can you answer my questions when I have them?". So, you're not just looking for one answer, you're looking for repeated action.

If this seems confusing, don't worry. This course will introduce the concept of perfective and imperfective verbs later.

«Отвеча́ешь» is a personal form, 'you answer'. Personal forms are used for the main verb of the sentence: «Ты мне отвеча́ешь?» 'Do you answer me?'

However, this sentence already has a main veb, «мо́жешь» 'you can'. Together with «мо́жешь», you need to use a different verb form, infinitive. Infinitive of «отвеча́ешь» is «отвеча́ть».

The difference between «отвеча́ть» and «отве́тить» is that:

• «Отве́тить» is a perfective verb. It refers to one, successful answer of the question. E.g. «Ты мо́жешь мне отве́тить?» 'Can you answer to me?' (I will just ask one question.)
• «Отвеча́ть» is an imperfective verb. It refers to answering during some period of time. «Ты мо́жешь мне отвеча́ть?» 'Can you be answering me?' (I will be asking questions during some period of time.)

It's perfectly correct Russian, but Duolingo expects you to be more literal:

• Вы мо́жете..? = Could/can you..?
• Вы не мо́жете..? = Couldn't/can't you..?

Of course, in real life those questions are interchangeable. But Duolingo doesn't allow such translations to make sure you know how to use negation in Russian.